Question: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{5z^2 - 40z + 35}{z^3 - z^2 - 42z}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {5(z^2 - 8z + 7)} {z(z^2 - z - 42)} $ $ x = \dfrac{5}{z} \cdot \dfrac{z^2 - 8z + 7}{z^2 - z - 42} $ Next factor the numerator and denominator. $ x = \dfrac{5}{z} \cdot \dfrac{(z - 7)(z - 1)}{(z - 7)(z + 6)}$ Assuming $z \neq 7$ , we can cancel the $z - 7$ $ x = \dfrac{5}{z} \cdot \dfrac{z - 1}{z + 6}$ Therefore: $ x = \dfrac{ 5(z - 1)}{ z(z + 6)}$, $z \neq 7$